"A punctiform source of light is standing inside a lake, at a height h of the surface. f is the fraction of the total of energy emitted that escapes directly from the lake, ignoring the light being absorbed in the water. Given n, the refractive index of water, determine f."

I understand that, since the maximum refraction angle is 90°, there is a maximum incident angle. The next image explains the principle:

The fraction of light that made out of the lake is a over the total circle, that is, 360°. So:

f = 2i/360 = i/180
i = arc sin (1/n)
f = (arc sin (1/n))/180

However, the answer I have for this exercise (and it does seem to be right, because it is from a University*) is $f = \tfrac12 - \tfrac{1}{2n} \sqrt{n^2 - 1}$. And I don't know what I did wrong. It is very important for me to solve this exercise, and I hope someone would have a hint of what I am doing wrong.

*It is a very old test (1969), and there is no resolution anywhere (just the final answer).

Second try, using Solid Angles:

At is the total Area of the light sphere of radius h:

At = 4 * pi * rt²
rt = h
At = 4 * pi * h²

Ap is the partial area of the circle of light that gets out of the water:

1 Answer
1

According to http://en.wikipedia.org/wiki/Solid_angle, the solid angle given by a cone with angle $2\theta$ is $\Omega=2\pi(1-\cos(\theta))$, i.e., it covers a fraction $\Omega/4\pi = (1-\cos\theta)/2$ of the sphere. This is exactly the fraction of the light leaving the water, where $\theta$ is just the angle you originally called $i$, i.e., $\theta=\sin^{-1}(1/n)$. Thus,
$$
f=(1-\cos(\sin^{-1}(1/n)))/2 = \tfrac12-\tfrac1{2n}\sqrt{n^2-1}\ .
$$

@LuanNico Although this is obviously the solution that was sought, since it gave the "correct" answer, I believe that this is not really the correct solution to the problem as it is formulated. This solution assumes that no light is reflected for the angles below total internal reflection. That is not true. The correct solution would require integrating expressions for transmittance derived from the Fresnel equations over the solid angle of the cone.
–
jkejApr 2 '13 at 15:12